Real-rooted atlas

A038291 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(3 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
131013
2960100169
32727090010002197
48110805400120001000028561
524340502700090000150000100000371293
6729145801215005400001350000180000010000004826809
72187510305103002835000945000018900000210000001000000062748517
8656117496020412001360800056700000151200000252000000240000000100000000815730721
9196835904907873200612360003061800001020600000226800000032400000002700000000100000000010604499373
1059049196830029524500262440000153090000061236000001701000000032400000000405000000003000000000010000000000137858491849
11177147649539010825650010825650007217100000336798000001122660000002673000000004455000000004950000000003300000000001000000000001792160394037
125314412125764038972340043302600003247695000017321040000067359600000019245600000004009500000000594000000000059400000000003600000000000100000000000023298085122481
13159432369087330138174660016888014000140733450000844400700000375289200000012509640000000312741000000005791500000000077220000000000702000000000003900000000000010000000000000302875106592253
144782969223205220483611310064481508000591080490000394053660000019702683000000750578400000002189187000000004864860000000008108100000000009828000000000008190000000000004200000000000001000000000000003937376385699289
Technical details
Node id
n-fe2b68140d990ced · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-1024c0779b1dc97d

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     3,     10;
  2:     9,     60,     100;
  3:    27,    270,     900,     1000;
  4:    81,   1080,    5400,    12000,     10000;
  5:   243,   4050,   27000,    90000,    150000,     100000;
  6:   729,  14580,  121500,   540000,   1350000,    1800000,    1000000;
  7:  2187,  51030,  510300,  2835000,   9450000,   18900000,   21000000,   10000000;
  8:  6561, 174960, 2041200, 13608000,  56700000,  151200000,  252000000,  240000000,  100000000;
  9: 19683, 590490, 7873200, 61236000, 306180000, 1020600000, 2268000000, 3240000000, 2700000000, 1000000000;
%F Formula
The row polynomials satisfy P(n,x) = (3+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3 + 10*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + 10*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.